Subgroups — #26029419

Order 11392.

346 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152132	C₂	1	1	yes	maximal, characteristic
2	#19151727	C₂	1	2	yes	maximal, complemented
3	#19151674	C₂	1	2	yes	maximal, complemented
4	#19151668	C₂	1	2	yes	maximal, complemented
5	#19151667	C₂	1	2	yes	maximal, complemented
6	#19151658	C₂	1	2	yes	maximal, complemented
7	#19151655	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151623	C₂	1	2	yes	maximal, complemented
9	#423×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13083557	C₂²	1	2	yes	—
11	C₂×C₈×D₈₉	C₂²	1	2	yes	—
12	#15668006	C₄	1	2	yes	—
13	#15668004	C₄	1	1	yes	characteristic
14	#15668000	C₄	1	1	yes	characteristic
15	#15667995	C₄	1	2	yes	—
16	#15667991	C₄	1	1	yes	characteristic
17	#15667986	C₄	1	1	yes	characteristic
18	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
19	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
20	C₄×C₂²⋊D₈₉	C₂²	1	2	yes	—
21	#15667969	C₂²	1	2	yes	—
22	C₄⋊#10623494	C₂²	1	1	yes	characteristic
23	C₄⋊#10623494	C₂²	1	1	yes	characteristic
24	#15667941	C₂²	1	2	yes	—
25	#15667941	C₂²	1	2	yes	—
26	C₂²⋊#10623494	C₂²	1	2	yes	—
27	#15667924	C₂²	1	2	yes	—
28	#15667924	C₂²	1	2	yes	—
29	#15667923	C₂²	1	2	yes	—
30	#15667922	C₂²	1	2	yes	—
31	C₈×Dic₈₉	C₂²	1	2	yes	—
32	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
33	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
36	C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
37	C₂×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
38	C₂×D₃₅₆	C₂×C₄	1	1	yes	characteristic
39	C₂×C₂²⋊D₈₉	C₂×C₄	1	2	yes	—
40	C₂×C₄×D₈₉	C₂³	1	2	yes	—
41	#13083557	—	4	2	no	—
42	C₂×#10623494	C₂³	1	2	yes	—
43	C₂×#10623494	C₂³	1	2	yes	—
44	C₈×D₈₉	—	4	2	no	—
45	C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
46	C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
47	#13083553	C₂×C₄	1	2	yes	—
48	#13083553	C₂×C₄	1	2	yes	—
49	#13083553	C₂×C₄	1	2	yes	—
50	#13083553	C₂×C₄	1	2	yes	—
51	C₂²⋊Dic₈₉	C₂×C₄	1	2	yes	—
52	#13083551	C₂×C₄	1	2	yes	—
53	#13083551	C₂×C₄	1	2	yes	—
54	#13083551	C₂×C₄	1	2	yes	—
55	#13083551	C₂×C₄	1	2	yes	—
56	C₄×Dic₈₉	C₂³	1	2	yes	—
57	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
60	M₁₆×C₈₉	—	4	1	no	—
61	M₁₆×C₈₉	—	4	1	no	—
62	C₂×C₈×C₈₉	C₂³	1	2	yes	—
63	C₂×C₈×C₈₉	C₂³	1	2	yes	—
64	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
65	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
66	C₄²×C₈₉	C₂³	1	1	yes	characteristic
67	C₄²×C₈₉	C₂³	1	1	yes	characteristic
68	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
69	C₂²×D₈₉	C₂²×C₄	1	2	yes	—
70	C₂×C₄×C₈₉	—	2	1	no	—
71	C₂×C₄×C₈₉	—	2	1	no	—
72	C₂×C₄×C₈₉	—	4	1	no	—
73	C₂×C₄×C₈₉	—	4	1	no	—
74	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
80	C₂²⋊D₈₉	—	4	2	no	—
81	C₂×Dic₈₉	C₂²×C₄	1	2	yes	—
82	C₂×Dic₈₉	C₂²×C₄	1	2	yes	—
83	C₂×Dic₈₉	C₂²×C₄	1	2	yes	—
84	D₃₅₆	—	4	1	no	—
85	C₄×D₈₉	—	2	2	no	—
86	C₄×D₈₉	—	2	2	no	—
87	Dic₁₇₈	—	4	1	no	—
88	C₈×C₈₉	—	2	2	no	—
89	C₈×C₈₉	—	2	2	no	—
90	#10623494	—	2	2	no	—
91	#10623494	—	2	2	no	—
92	C₂²×C₈₉	—	2	1	no	—
93	C₂²×C₈₉	—	2	1	no	—
94	C₂²×C₈₉	—	2	1	no	—
95	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
96	C₂×D₈₉	—	2	2	no	—
97	C₂×D₈₉	—	2	2	no	—
98	C₂×D₈₉	—	2	2	no	—
99	C₄×C₈₉	—	2	1	no	—
100	C₄×C₈₉	—	2	1	no	—
101	C₄×C₈₉	—	2	1	no	—
102	C₄×C₈₉	—	2	1	no	—
103	C₄×C₈₉	—	2	1	no	—
104	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
105	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
106	Dic₈₉	—	2	2	no	—
107	Dic₈₉	—	2	2	no	—
108	Dic₈₉	—	2	2	no	—
109	C₂×C₈₉	—	4	1	no	—
110	C₂×C₈₉	#520	1	1	yes	characteristic
111	C₂×C₈₉	#568	1	1	yes	characteristic
112	C₂×C₈₉	#568	1	1	yes	characteristic
113	D₈₉	—	4	2	no	—
114	#2995	—	89	1	no	maximal, Sylow
115	C₈₉	#2995	1	1	yes	characteristic, complemented, Sylow
116	#529	—	89	1	no	—
117	#446	—	89	2	no	—
118	#436	—	89	2	no	—
119	#435	—	89	2	no	—
120	#435	—	89	2	no	—
121	#433	—	89	2	no	—
122	#432	—	89	1	no	—
123	#424	—	89	2	no	—
124	#423	D₈₉	1	1	yes	characteristic, complemented
125	C₂×M₁₆	—	89	2	no	—
126	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
127	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
128	C₂²×C₈	—	89	2	no	—
129	#126	—	89	2	no	—
130	#125	—	89	1	no	—
131	#124	—	89	1	no	—
132	#123	—	89	2	no	—
133	C₂²⋊Q₈	—	89	1	no	—
134	#121	—	89	1	no	—
135	C₄×Q₈	—	89	1	no	—
136	C₄×D₄	—	89	2	no	—
137	C₄×D₄	—	89	1	no	—
138	#117	—	89	2	no	—
139	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
140	C₄⋊C₈	—	89	2	no	—
141	C₄⋊C₈	—	89	2	no	—
142	C₄⋊C₈	—	89	1	no	—
143	C₄⋊C₈	—	89	1	no	—
144	C₄⋊C₈	C₂×D₈₉	1	2	yes	—
145	C₄⋊C₈	C₂×D₈₉	1	2	yes	—
146	C₂²⋊C₈	—	89	2	no	—
147	C₂²⋊C₈	—	89	2	no	—
148	C₂²⋊C₈	—	89	2	no	—
149	#97	—	89	2	no	—
150	#97	—	89	2	no	—
151	C₄×C₈	—	89	2	no	—
152	C₂×Q₈	—	89	1	no	—
153	C₂×D₄	—	89	2	no	—
154	C₂×D₄	—	89	1	no	—
155	C₂²×C₄	—	89	2	no	—
156	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
157	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
158	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
159	M₁₆	—	4	1	no	—
160	M₁₆	—	4	1	no	—
161	M₁₆	—	356	2	no	—
162	C₂×C₈	—	89	2	no	—
163	C₂×C₈	—	89	2	no	—
164	C₂×C₈	—	356	2	no	—
165	C₂×C₈	C₂²×D₈₉	1	2	yes	—
166	C₂×C₈	C₂²×D₈₉	1	2	yes	—
167	C₄⋊C₄	—	89	2	no	—
168	C₄⋊C₄	—	89	2	no	—
169	C₄⋊C₄	—	89	1	no	—
170	C₄⋊C₄	—	89	2	no	—
171	C₄⋊C₄	—	89	1	no	—
172	C₄⋊C₄	—	89	2	no	—
173	C₂²⋊C₄	—	89	2	no	—
174	C₂²⋊C₄	—	89	2	no	—
175	C₂²⋊C₄	—	89	2	no	—
176	C₂²⋊C₄	—	89	2	no	—
177	C₂²⋊C₄	—	89	2	no	—
178	C₄²	—	89	2	no	—
179	C₄²	C₄×D₈₉	1	1	yes	characteristic
180	C₄²	C₄×D₈₉	1	1	yes	characteristic
181	C₄²	C₂²×D₈₉	1	1	yes	characteristic
182	C₄²	C₂²×D₈₉	1	1	yes	characteristic
183	C₂³	—	89	2	no	—
184	C₂³	C₂×C₄×D₈₉	1	1	yes	characteristic
185	Q₈	—	356	1	no	—
186	D₄	—	356	1	no	—
187	D₄	—	356	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	4	1	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	—	89	2	no	—
193	C₂×C₄	—	89	2	no	—
194	C₂×C₄	—	89	2	no	—
195	C₂×C₄	—	178	2	no	—
196	C₂×C₄	—	178	2	no	—
197	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
199	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
200	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
201	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
202	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
203	C₈	—	2	2	no	—
204	C₈	—	2	2	no	—
205	C₈	—	178	2	no	—
206	C₈	—	178	2	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	178	2	no	—
211	C₂²	—	178	2	no	—
212	C₂²	—	178	2	no	—
213	C₂²	C₂²×C₄×D₈₉	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	178	2	no	—
220	C₄	—	178	2	no	—
221	C₄	—	178	2	no	—
222	C₄	#15668057	1	1	yes	characteristic
223	C₄	#15668060	1	1	yes	characteristic
224	C₂	—	4	1	no	—
225	C₂	—	356	2	no	—
226	C₂	#19152079	1	1	yes	characteristic
227	C₂	#19152364	1	1	yes	characteristic
228	C₂	#19152364	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₄
Fitting	#423×C₈₉	C₂
prime radical	#15667986	C₄
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	#423	D₈₉
hypocenter	C₈₉	#2995
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #19151727	C₂	semidirect
3. #19151674	C₂	semidirect
4. #19151668	C₂	semidirect
5. #19151667	C₂	semidirect
6. #19151658	C₂	semidirect
7. #19151655	C₂	semidirect
8. #19151623	C₂	semidirect
9. #423×C₈₉	C₂	semidirect
115. C₈₉	#2995	semidirect
124. #423	D₈₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#19152132	2
#19151727	2
#19151674	2
#19151668	2
#19151667	2
#19151658	2
#19151655	2
#19151623	2
#423×C₈₉	2
#2995	89